+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
-(* ||A|| E.Tassi, S.Zacchiroli *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU Lesser General Public License Version 2.1 *)
-(* *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/Z/plus".
-
-include "Z/z.ma".
-include "nat/minus.ma".
-
-definition Zplus :Z \to Z \to Z \def
-\lambda x,y.
- match x with
- [ OZ \Rightarrow y
- | (pos m) \Rightarrow
- match y with
- [ OZ \Rightarrow x
- | (pos n) \Rightarrow (pos (pred ((S m)+(S n))))
- | (neg n) \Rightarrow
- match nat_compare m n with
- [ LT \Rightarrow (neg (pred (n-m)))
- | EQ \Rightarrow OZ
- | GT \Rightarrow (pos (pred (m-n)))] ]
- | (neg m) \Rightarrow
- match y with
- [ OZ \Rightarrow x
- | (pos n) \Rightarrow
- match nat_compare m n with
- [ LT \Rightarrow (pos (pred (n-m)))
- | EQ \Rightarrow OZ
- | GT \Rightarrow (neg (pred (m-n)))]
- | (neg n) \Rightarrow (neg (pred ((S m)+(S n))))] ].
-
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "integer plus" 'plus x y = (cic:/matita/Z/plus/Zplus.con x y).
-
-theorem Zplus_z_OZ: \forall z:Z. z+OZ = z.
-intro.elim z.
-simplify.reflexivity.
-simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-(* theorem symmetric_Zplus: symmetric Z Zplus. *)
-
-theorem sym_Zplus : \forall x,y:Z. x+y = y+x.
-intros.elim x.rewrite > Zplus_z_OZ.reflexivity.
-elim y.simplify.reflexivity.
-simplify.
-rewrite < plus_n_Sm. rewrite < plus_n_Sm.rewrite < sym_plus.reflexivity.
-simplify.
-rewrite > nat_compare_n_m_m_n.
-simplify.elim nat_compare.simplify.reflexivity.
-simplify. reflexivity.
-simplify. reflexivity.
-elim y.simplify.reflexivity.
-simplify.rewrite > nat_compare_n_m_m_n.
-simplify.elim nat_compare.simplify.reflexivity.
-simplify. reflexivity.
-simplify. reflexivity.
-simplify.rewrite < plus_n_Sm. rewrite < plus_n_Sm.rewrite < sym_plus.reflexivity.
-qed.
-
-theorem Zpred_Zplus_neg_O : \forall z:Z. Zpred z = (neg O)+z.
-intros.elim z.
- simplify.reflexivity.
- elim n.
- simplify.reflexivity.
- simplify.reflexivity.
- simplify.reflexivity.
-qed.
-
-theorem Zsucc_Zplus_pos_O : \forall z:Z. Zsucc z = (pos O)+z.
-intros.elim z.
- simplify.reflexivity.
- simplify.reflexivity.
- elim n.
- simplify.reflexivity.
- simplify.reflexivity.
-qed.
-
-theorem Zplus_pos_pos:
-\forall n,m. (pos n)+(pos m) = (Zsucc (pos n))+(Zpred (pos m)).
-intros.
-elim n.elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-elim m.
-simplify.rewrite < plus_n_Sm.
-rewrite < plus_n_O.reflexivity.
-simplify.rewrite < plus_n_Sm.
-rewrite < plus_n_Sm.reflexivity.
-qed.
-
-theorem Zplus_pos_neg:
-\forall n,m. (pos n)+(neg m) = (Zsucc (pos n))+(Zpred (neg m)).
-intros.reflexivity.
-qed.
-
-theorem Zplus_neg_pos :
-\forall n,m. (neg n)+(pos m) = (Zsucc (neg n))+(Zpred (pos m)).
-intros.
-elim n.elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-theorem Zplus_neg_neg:
-\forall n,m. (neg n)+(neg m) = (Zsucc (neg n))+(Zpred (neg m)).
-intros.
-elim n.elim m.
-simplify.reflexivity.
-simplify.reflexivity.
-elim m.
-simplify.rewrite > plus_n_Sm.reflexivity.
-simplify.rewrite > plus_n_Sm.reflexivity.
-qed.
-
-theorem Zplus_Zsucc_Zpred:
-\forall x,y. x+y = (Zsucc x)+(Zpred y).
-intros.elim x.
- elim y.
- simplify.reflexivity.
- rewrite < Zsucc_Zplus_pos_O.rewrite > Zsucc_Zpred.reflexivity.
- simplify.reflexivity.
- elim y.
- simplify.reflexivity.
- apply Zplus_pos_pos.
- apply Zplus_pos_neg.
- elim y.
- rewrite < sym_Zplus.rewrite < (sym_Zplus (Zpred OZ)).
- rewrite < Zpred_Zplus_neg_O.rewrite > Zpred_Zsucc.simplify.reflexivity.
- apply Zplus_neg_pos.
- rewrite < Zplus_neg_neg.reflexivity.
-qed.
-
-theorem Zplus_Zsucc_pos_pos :
-\forall n,m. (Zsucc (pos n))+(pos m) = Zsucc ((pos n)+(pos m)).
-intros.reflexivity.
-qed.
-
-theorem Zplus_Zsucc_pos_neg:
-\forall n,m. (Zsucc (pos n))+(neg m) = (Zsucc ((pos n)+(neg m))).
-intros.
-apply (nat_elim2
-(\lambda n,m. (Zsucc (pos n))+(neg m) = (Zsucc ((pos n)+(neg m))))).intro.
-intros.elim n1.
-simplify. reflexivity.
-elim n2.simplify. reflexivity.
-simplify. reflexivity.
-intros. elim n1.
-simplify. reflexivity.
-simplify.reflexivity.
-intros.
-rewrite < (Zplus_pos_neg ? m1).
-elim H.reflexivity.
-qed.
-
-theorem Zplus_Zsucc_neg_neg :
-\forall n,m. Zsucc (neg n) + neg m = Zsucc (neg n + neg m).
-intros.
-apply (nat_elim2
-(\lambda n,m. Zsucc (neg n) + neg m = Zsucc (neg n + neg m))).intro.
-intros.elim n1.
-simplify. reflexivity.
-elim n2.simplify. reflexivity.
-simplify. reflexivity.
-intros. elim n1.
-simplify. reflexivity.
-simplify.reflexivity.
-intros.
-rewrite < (Zplus_neg_neg ? m1).
-reflexivity.
-qed.
-
-theorem Zplus_Zsucc_neg_pos:
-\forall n,m. Zsucc (neg n)+(pos m) = Zsucc ((neg n)+(pos m)).
-intros.
-apply (nat_elim2
-(\lambda n,m. Zsucc (neg n) + (pos m) = Zsucc (neg n + pos m))).
-intros.elim n1.
-simplify. reflexivity.
-elim n2.simplify. reflexivity.
-simplify. reflexivity.
-intros. elim n1.
-simplify. reflexivity.
-simplify.reflexivity.
-intros.
-rewrite < H.
-rewrite < (Zplus_neg_pos ? (S m1)).
-reflexivity.
-qed.
-
-theorem Zplus_Zsucc : \forall x,y:Z. (Zsucc x)+y = Zsucc (x+y).
-intros.elim x.
- elim y.
- simplify. reflexivity.
- simplify.reflexivity.
- rewrite < Zsucc_Zplus_pos_O.reflexivity.
- elim y.
- rewrite < (sym_Zplus OZ).reflexivity.
- apply Zplus_Zsucc_pos_pos.
- apply Zplus_Zsucc_pos_neg.
- elim y.
- rewrite < sym_Zplus.rewrite < (sym_Zplus OZ).simplify.reflexivity.
- apply Zplus_Zsucc_neg_pos.
- apply Zplus_Zsucc_neg_neg.
-qed.
-
-theorem Zplus_Zpred: \forall x,y:Z. (Zpred x)+y = Zpred (x+y).
-intros.
-cut (Zpred (x+y) = Zpred ((Zsucc (Zpred x))+y)).
-rewrite > Hcut.
-rewrite > Zplus_Zsucc.
-rewrite > Zpred_Zsucc.
-reflexivity.
-rewrite > Zsucc_Zpred.
-reflexivity.
-qed.
-
-
-theorem associative_Zplus: associative Z Zplus.
-change with (\forall x,y,z:Z. (x + y) + z = x + (y + z)).
-(* simplify. *)
-intros.elim x.
- simplify.reflexivity.
- elim n.
- rewrite < Zsucc_Zplus_pos_O.rewrite < Zsucc_Zplus_pos_O.
- rewrite > Zplus_Zsucc.reflexivity.
- rewrite > (Zplus_Zsucc (pos n1)).rewrite > (Zplus_Zsucc (pos n1)).
- rewrite > (Zplus_Zsucc ((pos n1)+y)).apply eq_f.assumption.
- elim n.
- rewrite < (Zpred_Zplus_neg_O (y+z)).rewrite < (Zpred_Zplus_neg_O y).
- rewrite < Zplus_Zpred.reflexivity.
- rewrite > (Zplus_Zpred (neg n1)).rewrite > (Zplus_Zpred (neg n1)).
- rewrite > (Zplus_Zpred ((neg n1)+y)).apply eq_f.assumption.
-qed.
-
-variant assoc_Zplus : \forall x,y,z:Z. (x+y)+z = x+(y+z)
-\def associative_Zplus.
-
-(* Zopp *)
-definition Zopp : Z \to Z \def
-\lambda x:Z. match x with
-[ OZ \Rightarrow OZ
-| (pos n) \Rightarrow (neg n)
-| (neg n) \Rightarrow (pos n) ].
-
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "integer unary minus" 'uminus x = (cic:/matita/Z/plus/Zopp.con x).
-
-theorem Zopp_Zplus: \forall x,y:Z. -(x+y) = -x + -y.
-intros.
-elim x.elim y.
-simplify. reflexivity.
-simplify. reflexivity.
-simplify. reflexivity.
-elim y.
-simplify. reflexivity.
-simplify. reflexivity.
-simplify. apply nat_compare_elim.
-intro.simplify.reflexivity.
-intro.simplify.reflexivity.
-intro.simplify.reflexivity.
-elim y.
-simplify. reflexivity.
-simplify. apply nat_compare_elim.
-intro.simplify.reflexivity.
-intro.simplify.reflexivity.
-intro.simplify.reflexivity.
-simplify.reflexivity.
-qed.
-
-theorem Zopp_Zopp: \forall x:Z. --x = x.
-intro. elim x.
-reflexivity.reflexivity.reflexivity.
-qed.
-
-theorem Zplus_Zopp: \forall x:Z. x+ -x = OZ.
-intro.elim x.
-apply refl_eq.
-simplify.
-rewrite > nat_compare_n_n.
-simplify.apply refl_eq.
-simplify.
-rewrite > nat_compare_n_n.
-simplify.apply refl_eq.
-qed.
-
projects / helm.git / blobdiff